Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sradrng | Structured version Visualization version GIF version |
Description: Condition for a subring algebra to be a division ring. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
sraring.1 | ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) |
sraring.2 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
sradrng | ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 19728 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | sraring.1 | . . . 4 ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) | |
3 | sraring.2 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | sraring 31340 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) |
5 | 1, 4 | sylan 583 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) |
6 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2736 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
8 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 6, 7, 8 | isdrng 19725 | . . . . 5 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))) |
10 | 9 | simprbi 500 | . . . 4 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})) |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})) |
12 | eqidd 2737 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (Base‘𝑅) = (Base‘𝑅)) | |
13 | 2 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 = ((subringAlg ‘𝑅)‘𝑉)) |
14 | simpr 488 | . . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ 𝐵) | |
15 | 14, 3 | sseqtrdi 3937 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ (Base‘𝑅)) |
16 | 13, 15 | srabase 20169 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (Base‘𝑅) = (Base‘𝐴)) |
17 | 13, 15 | sramulr 20171 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (.r‘𝑅) = (.r‘𝐴)) |
18 | 17 | oveqdr 7219 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
19 | 12, 16, 18 | unitpropd 19669 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (Unit‘𝑅) = (Unit‘𝐴)) |
20 | eqidd 2737 | . . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (0g‘𝑅) = (0g‘𝑅)) | |
21 | 13, 20, 15 | sralmod0 20179 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (0g‘𝑅) = (0g‘𝐴)) |
22 | 21 | sneqd 4539 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → {(0g‘𝑅)} = {(0g‘𝐴)}) |
23 | 16, 22 | difeq12d 4024 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = ((Base‘𝐴) ∖ {(0g‘𝐴)})) |
24 | 11, 19, 23 | 3eqtr3d 2779 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0g‘𝐴)})) |
25 | eqid 2736 | . . 3 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
26 | eqid 2736 | . . 3 ⊢ (Unit‘𝐴) = (Unit‘𝐴) | |
27 | eqid 2736 | . . 3 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
28 | 25, 26, 27 | isdrng 19725 | . 2 ⊢ (𝐴 ∈ DivRing ↔ (𝐴 ∈ Ring ∧ (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0g‘𝐴)}))) |
29 | 5, 24, 28 | sylanbrc 586 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 ‘cfv 6358 Basecbs 16666 .rcmulr 16750 0gc0g 16898 Ringcrg 19516 Unitcui 19611 DivRingcdr 19721 subringAlg csra 20159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-mgp 19459 df-ur 19471 df-ring 19518 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-drng 19723 df-sra 20163 |
This theorem is referenced by: rgmoddim 31361 extdggt0 31400 |
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